Multicriteria Assessment Method for Network Structure Congestion Based on Traffic Data Using Advanced Computer Vision

Abstract

Overloading of network structures is a problem that we encounter every day in many areas of life. The most associative structure is the transport graph. In many megacities around the world, the so-called intelligent transport system (ITS) is successfully operating, allowing real-time monitoring and making changes to traffic management while choosing the most effective solutions. Thanks to the emergence of more powerful computing resources, it has become possible to build more complex and realistic mathematical models of traffic flows, which take into account the interactions of drivers with road signs, markings, and traffic lights, as well as with each other. Simulations using high-performance systems can cover road networks at the scale of an entire city or even a country. It is important to note that the tool being developed is applicable to most network structures described by such mathematical apparatuses as graph theory and the applied theory of network planning and management that are widely used for representing the processes of organizing production and enterprise management. The result of this work is a developed model that implements methods for modeling the behavior of traffic flows based on physical modeling and machine learning algorithms. Moreover, a computer vision system is proposed for analyzing traffic on the roads, which, based on vision transformer technologies, provides high accuracy in detecting cars, and using optical flow, allows for significantly faster processing. The accuracy is above 90% with a processing speed of more than ten frames per second on a single video card.

1. Introduction

The development of information and communication technologies built on a network principle has led to the widespread dissemination of network structures in society. The concepts of “network structures”, “social network” and “network culture” have firmly entered scientific circulation, but their content is often blurred. It must be emphasized that, in the most general form, the concepts of “network” and “network structure” are used in modern scientific knowledge to denote an interacting set of objects connected to each other by communication lines. Network problems associated with the construction of macrosystems based on the combination of several microsystems (subsystems) using graph theory are of great relevance in various fields of application. These include the task of creating complex microprocessor systems, the task of building distributed network connections based on combining several autonomous computer networks into a single network, and also the task of building transport systems that include heterogeneous transport networks that have different topologies and use different types of transport. A similar problem arises when creating complex chemical compounds whose molecular formulas are well described graphically. The mathematical objects in such problems are graphs and their matrix representations. An optimal combination of mathematical objects is possible when using algorithms based on the theory and methods of combinatorial analysis. The difficulty in solving such problems lies in the computational complexity, which, as the number and size of graphs grow, quickly becomes unacceptably high and requires too much computer memory.

Overloading of network structures is a problem that we, participants in such structures, face every day in many areas of modern life. Perhaps the most associative network structure is the transport graph. In many megapolises around the world, the so-called intelligent transport system (ITS) is successfully operating and allowing real-time monitoring in addition to making changes to traffic management and choosing the most effective solutions. Through the emergence of more powerful computing resources, it has become possible to build more complex and realistic mathematical models of traffic flows that take into account the interactions of drivers with road signs, markings, traffic lights and with each other. Simulations using high-performance systems can cover the road network at the scale of an entire city or even a country. The emergence in recent years of new vehicle flow data, collected by sensors, CCTV cameras and other smart devices, opens up new opportunities for studying the flow patterns of speakers, and also allows for the precise calibration of models for use in specific locations, which can improve predictions obtained in real time within the framework of computational experiments. All of the above allows us to draw conclusions about the relevance and prospects of the chosen area of research.

The goal of this research is to develop a universal tool based on mathematical models that allows the aggregation of various criteria for assessing network structures to effectively support modeling-based decision making. To find effective management strategies and rational solutions for the design and organization of network structures, it is necessary to take into account a wide range of characteristics and patterns of influence of external and internal factors. Theories on the problems of network structures have been developed by researchers in various fields—physicists, mathematicians, operations research specialists, logisticians and economists, etc. Extensive experience has been accumulated through studying motion processes. However, the general level of research and its practical usage is currently insufficient due to the following factors:

• the flow of the network structure is unstable and varied, and obtaining objective information is the most complex and resource-intensive element of the control system;
• management-quality criteria are contradictory, and it is necessary to ensure uninterrupted traffic in the network structure while simultaneously reducing damage and introducing restrictions on speed and direction;
• criteria for evaluating modeling results depend on preferences (the weights of some criteria depend on the values of other criteria) and the workload of the network structure at a particular point in time;
• external parameters influencing the overall result, for example, weather and climatic conditions affecting the condition of roads;
• when taking into account the nature of the motion process in network structures, the implementation of motion-control decisions cannot have an absolutely accurate result from mathematical modeling due to the peculiarities of the motion process for all elements of the system, which leads to unforeseen effects;
• the effectiveness of solutions is assessed using a multidimensional vector criterion, and manual analysis of solutions is difficult without the use of a computer.

Thus, the difficulties of formalizing the process of movement of network structures taking into account the vector criterion have become serious reasons for the lag in the results of scientific research from the requirements of practice [1,2,3,4,5,6]. These intelligent systems are now the way for traffic monitoring and prediction, so it will be possible to make routes optimized.

2. Multicriteria Assessment Methods

Intelligent analysis requires different smart algorithms. One area for applying such analysis is multicriteria assessment. It is widely used in different tasks, such as manufacturing [7] and sustainability [8]. In the transportation sector, this approach has also caught on. The authors of [9] have developed a system of evaluation of nine parameters to improve the quality of passenger service. Thanks to the method of comprehensive evaluation, it is possible to achieve a positive response. And in [10], multicriteria evaluations are proposed for metro transportation. Thanks to such an algorithm, it is possible to predict locations for new stations and improve the quality of life of the population. Let us see this approach in more detail. The multicriteria approach for traffic is justified because of the many factors and constraints that can influence the selection of the most efficient route or path for travel: environmental aspects, travel time, cost, etc.

In most developed countries, network structures are increasingly used in various subject areas, and interest in network models in management is growing. Works have been published on general issues of network mathematics, network methods of decision making, social networks and network-centric control in multi-agent systems, and numerous examples of technological networks have been considered. The structure of the network is modeled by a graph, the vertices of which correspond to autonomous functional units called agents, and the edges reflect interactions between agents. The interaction is that one of the agents sends a message to another and receives a response from him. Such an interaction can be initiated by any of a pair of agents and repeated many times, with each subsequent interaction only possible after the previous one has been completed. The rules for constructing network models are determined by the scope of their application, including network planning and management, as well as the corresponding software of modern computers developed for constructing and analyzing networks. Based on a number of classic works and publications in recent years, network structures are compared with the following:

• Hierarchical (vertical) structures with a single control center. In the world of biosystems and in human society, hierarchical structures are based on relationships of dominance–subordination (subordination) between elements of higher and lower ranks, in contrast to network structures, which are based on a tendency towards equalization of ranks and cooperation. As for human society, hierarchical structures are characteristic of both traditional societies (say, the relationship between lords and vassals in the era of feudalism) and bureaucracies, which are modeled on many modern political, cultural and scientific institutions.
• Quasi-market structures in which competition between elements prevails over cooperation between them. Market systems are based on element autonomy, equivalent exchange and competitive relations. Similar to the market structures of human society, many animals, including fish, insects, crustaceans, etc., have competition quasi-markets in biosystems, which can include analogues of chains of contracts, transactions between suppliers, resellers and consumers so-called metabolic chains [11,12].

Figure 1 shows network structures and their architecture.

The development of modern approaches to managing transport processes is associated with the integration of transport systems into a complex of mobile services for the delivery of goods and passengers by road. The practice of implementing logistics approaches in flow management shows the effectiveness of an integrated perception of traffic problems and flow dynamics, both in the sphere of distribution of goods and passengers and in transport itself. Engineering, technology and transport economics today are filled with ideas of network structures and digital components from the standpoint of informatization of transport process management and financial support of contractual and legal relations. Analogies introduced into the theory of economic teachings, classical transport science and other scientific areas, in fact, determined the current state and prospects for the development of these areas of research. They served as the basis for the development of the network economy and supply chain management. Biology and its forms of organization have become the prototype of a network form of interaction between objects in a competitive market environment. Psychology and intelligence served as the basis for the development of cognitive control models and the development of basic principles for managing transport systems. These approaches defined the procedure for constructing intelligent forecasting and control models for systems of a high level of complexity. Transport systems and transport processes are no exception to the increasing degree of complexity; therefore, the search for approaches, models and methods in these conditions is a relevant and necessary condition for the implementation of the most ambitious plans. The creation of intelligent transport systems today is one of the key areas for managing transport processes, the dynamics of transport flows, the demand for transport services and, in general, the entire complex of delivery operations in supply chains. The peculiarities of production and consumption markets leave their marks on the forms of organization and methods of interaction in conditions of de-centralization and the individualization of products and services, including logistics and transport. A characteristic feature of today is the emergence of network, and in the future, “cloud” production. Such technologies require the development of modern approaches to organization and planning, which are subject to a shift in emphasis from automation of operations to automation of management, including all stages of the life cycle [12,13,14,15].

Transport networks are typical complex network structures. Currently, a lot of literature is devoted to the study and modeling of traffic flows. Modeling of traffic flows includes a whole range of works aimed at studying the situation on the roads, including indicating problem areas. When compiling a mathematical model, not only the actual movement of transport over time is taken into account, but also future congestion. Transport modeling allows you to calculate the average speed on a complex section and propose solutions to increase traffic volumes. The models used to analyze transport networks are very diverse. At the moment, there is no comprehensive classification of modeling methods. Systematization, depending on the tasks to be solved, was carried out according to different criteria [16]. In recent years, new methods for optimizing transport routes have become possible, allowing calculation of the most efficient routes based on large amounts of data such as road conditions, vehicles, seasonality and delivery schedules. Examples of the application of optimization algorithms in a transport network include the following:

• taxi platforms such as Uber, Lyft and Gett use artificial intelligence to determine rational routes, manage prices and determine the arrival time for each trip. Algorithms analyze data about traffic, weather, time of day, congestion, roadworks and other road conditions, as well as data about orders and drivers to suggest the most efficient route for each order.
• Logistics companies such as DHL and FedEx use artificial intelligence to analyze data about cargo, transport, routes and delivery conditions to optimize vehicle routes, which allows the reduction cargo delivery time and the reduction transportation costs.
• Electric vehicle manufacturer Tesla uses proprietary artificial intelligence algorithms to optimize routes and predict the battery charges of electric vehicles. Tesla uses data about the distance to the nearest charging station, driving speed, time of day and other parameters to suggest an efficient charging route for each electric vehicle. This makes it possible to increase the efficiency of using electric vehicles and reduce their charging time. Traditional automakers such as Audi also have their own designs.
• Urban transport companies use artificial intelligence to optimize bus routes, taking into account factors such as traffic jams, passenger flow and traffic schedules.

There are several areas for optimizing transport routes, including:

• Deep learning algorithms that allow users to analyze large amounts of data and predict optimal routes.
• Genetic algorithms that mimic the process of natural selection to find the best solutions to a particular problem.
• Clustering methods that group data into different clusters to find the most efficient route for each vehicle in each cluster.
• Evolutionary algorithms that use the principles of natural selection and mutation to find the best solution to a particular problem.
• Dynamic programming methods that take into account all possible route combinations and select a rational route for each vehicle.
• Multi-agent modeling methods use agents representing different vehicles to find rational routes in real time.

One way to simplify the problem of optimizing transport routes is to normalize the priority levels of various parameters. Typically, normalization means scaling parameter values from zero to one. Additionally, the normalization results of each parameter are converted into a value based on the priority level. Decision making must simultaneously consider economic, social and environmental impacts over the long term. The complexity of decision making lies in the participation of many stakeholders pursuing different goals, the influence of many factors, the presence of a wide variety of management decisions at different levels of management, a large number of evaluation criteria and the complexity of calculations. The need to take into account these limitations has led to the active development of the scientific field of multicriteria decision analysis (MCDA) or multicriteria decision making (MCDM). MCDM methods are an important part of the theory of decision making and analysis, the main goal of which is to solve four types of problems: selecting the best solution from a set, ranking and sorting alternatives, describing and systematizing decisions and, lastly, the consequences of their adoption. Adoption and implementation are used for evaluation and further management. Currently, MCDM methods are actively used in the fields of climate change, sustainable development, economics, engineering, supply chain management, energy consumption, reverse logistics and corporate sustainability (Figure 2). MCDM methods are divided into two categories: MODM—multicriteria decision making and MADM—multi-attribute decision making. MODM models include a large number of alternative solutions, and the goal of considering a problem is to determine the optimal alternative, given a set of well-defined constraints, by solving a mathematical model. MADM models are discrete and are used for ranking, in which a finite number of proposed alternatives are evaluated against various weighted attributes to produce a preference rating that describes the effectiveness of each alternative in achieving the attribute’s goal. To improve the estimation efficiency, MCDM models can be combined with fuzzy set theory, rough set theory, gray set theory, etc. The disadvantages of using MCDM models include the difficulty of data collection, poor initial information and the increased complexity of the decision-making process. The works of various authors note that there are no universal MCDM methods suitable for all decision-making situations, which leads to the problem of choice [17,18,19,20,21,22,23,24,25].

MCDA stands for Multiple Criteria Decision Analysis, whereas MCDM stands for Multiple Criteria Decision Making. Both terms refer to a field of study and a set of methods and techniques that aim to support decision-making processes involving multiple conflicting criteria and objectives.

MCDA/MCDM focuses on situations where decision makers need to evaluate and compare several alternatives based on multiple criteria, which may have different weights or priorities. These methods provide a structured approach to systematically analyze, assess, and rank alternatives, while taking into account various decision criteria and preferences.

The general process of MCDA/MCDM involves the following steps:

• Clearly define the decision problem, identify the alternatives, and specify the relevant criteria to be considered.
• Establish a set of criteria that are meaningful and relevant to the decision problem. These criteria can be quantitative (e.g., cost, time) or qualitative (e.g., environmental impact, user satisfaction).
• Determine the relative importance or weights for each of the criteria based on the decision maker’s preferences. This step reflects the different priorities assigned to the criteria.
• Gather data related to the criteria and analyze them to derive meaningful information and insights. This may involve statistical analysis, modeling, or other quantitative techniques.
• Alternative Evaluation and Scoring: Evaluate each alternative against the established criteria, assigning scores or values based on how well they perform for each criterion. This can be achieved using numerical scales, pairwise comparisons, or other elicitation methods.
• Combine the scores or values from step 5, applying weighting to generate an overall aggregated assessment of the alternatives. This step quantitatively compares and ranks the alternatives based on their performance across multiple criteria.
• Assess the robustness and sensitivity of the results by testing different scenarios, varying criteria weights, or considering uncertainty in the data or preferences.

MCDA/MCDM methods include a range of approaches such as the Analytic Hierarchy Process (AHP), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) and ELECTRE (Elimination and Choice Translating Reality). Each method has its own characteristics, strengths, and weaknesses, making them suitable for different decision contexts.

Overall, the goal of MCDA/MCDM is to provide decision support by structuring and managing complex decision problems with multiple criteria, enabling decision makers to make more informed and objective decisions by considering the trade-offs and preferences associated with the various criteria.

The most commonly used MCDM methods in the field of logistics and supply chain management are as follows: AHP—analytical hierarchical process, ANP—analytical network process, TOPSIS—method of ordered preference through similarity to the ideal solution, DEMATEL—method of testing and evaluating solutions, ELECTR—exclusion and selection in reality, PROMETHEE—method for organizing preference sorting to evaluate alternatives, VIKOR—multicriteria optimization and compromise solution. The decision-making process using MCDM includes three main stages illustrated in Figure 3.

Analysis of the literature showed that AHP and TOPSIS are most often used. A characteristic feature of the first is that it allows you to compare selected criteria and build a comparison matrix. With its help, at the next stage, global and local preferences are determined and the correspondence coefficient is calculated. At the final stage of evaluation, a final rating of acceptable alternatives is compiled based on the calculation of their utility function. TOPSIS comes down to measuring similarity to an ideal solution and is a classification method based on the degree of proximity: alternative options are arranged on a scale of agreement with the ideal and anti-ideal. The results are ranked based on the weight of the criteria used within the standard procedure. The best solution is considered to be the one that is closest to the ideal or furthest from it. The integral indicator obtained in this way determines the position of one or another option in the ranking. The AHP method is rarely used when there is a large set of criteria. Thus, with 24 criteria the matrix will include 24 columns and rows whereas usually their number does not exceed 10. In addition, the AHP method often involves subjective determination of the weight of the criteria depending on the opinion of experts. The interdependence of alternatives and criteria also gives rise to problems that can lead to inconsistency of the evaluated options with the ranking criteria and rating inversion. This is why, for example, the TOPSIS method is used to select alternative routes [26,27,28,29].

Based on the results of the publications review, 58 articles were selected using multicriteria evaluation methods. The distribution of these works by area of application is shown in

Table 1. Article distribution across application areas.

Application Area	Number of Publications	Proportion
Auto transport	12	20.69%
Public transport	11	18.97%
Logistics	10	17.24%
Air transport	8	13.79%
Maritime transport	6	10.34%
Rail transport	4	6.90%
Transport network planning	4	6.90%
Transport operations	2	3.45%
Pedestrian transport network	1	1.72%
Total	58	100%

.

To finalize the selection of criteria, various parameters from transport studies around the world were taken into account. The selected criteria are based on articles in which vehicle routing is a key objective, as well as studies that define criteria related to vehicle performance. The information is summarized in

Table 2. The most common parameters influencing route selection.

	Reference	[30]	[31]	[32]	[33]	[34]	[35]	[36]	[37]
Criteria
Distance			x		x		x	x
Geometric characteristics		x			x				x
Energy consumption				x
Road surface quality		x				x
Number of passengers						x
Travel time			x		x			x
Travel cost									x
Traffic flow		x	x		x		x		x
Availability of up-to-date data		x				x		x
Road type						x
Weather conditions				x
Environmental friendliness							x
Number of stops									x
Safety		x		x				x

.

The most commonly used parameters in research are technical criteria: distance, geometric characteristics of the transport network and travel time. Economic criteria include the cost of travel on a toll route, although this is quite rare in the transport network. Location criteria include the congestion of the transport network, the quality of the road surface and the availability of up-to-date route data. The safety criterion, in turn, includes sub-criteria that take into account statistical data: the number of road users, violations, injuries, deaths and other criteria. The criteria are divided into qualitative and quantitative criteria. Quantitative criteria include: distance, travel time, workload, cost of travel and safety. Qualitative criteria are determined subjectively based on expert opinions and relate to features or objects of the road [38,39,40,41,42,43,44].

3. Mathematical Models for Searching for Rational Alternatives

Mathematical modeling, due to its interpretability, remains one of the most popular methods for simulating various data and practicing algorithms. Traffic modeling is presented in [45]. The authors present the main mathematical models for road traffic and examples of traffic congestion calculations for different road sections. In [46], hybrid algorithms of convolutional and recurrent networks were proposed to improve the accuracy of traffic prediction and bypass other known models.

To classify the routes, the characteristics of each segment of the road network were calculated based on various data sources described in detail below. Accurate assessment of traffic conditions (volume, density and speed) is critical for traffic assessment, management and forecasting. Traditionally, relevant information about the road network is collected using various data sources, including fixed transmitters, surveillance cameras and other devices, as well as GPS sensors installed in the vehicle. TkStar TK-816 is a GPS tracker that works on existing GSM/GPRS networks and GPS satellites; this product can detect and monitor any remote targets via SMS, app and internet. It utilizes the most advanced dual positioning technology of GPS and AGPS. This device can directly connect to the OBD port of the vehicle. This tracker allows easy interaction with the device via the Winnes GPS mobile app for Android and iOS. The app allows users to conveniently configure the tracker as well as track the location of the monitored object in real time.

However, covering the entire road network with such equipment is expensive, and areas without data cannot be well surveyed. Additionally, fixed data sources can be damaged due to hardware or software failures, network communications and maintenance. Thus, solving the problem of a lack of data is one of the most important tasks for subsequent mathematical modeling of road network congestion.

Many countries use stationary traffic sensors that measure speed and traffic density on key highways as a good source of clean traffic data but covering even one megapolis would require huge investment. To do this, it is necessary to collect data on street congestion from various types of devices, and to process, analyze and display them on a map. Processing different types of data is expensive in terms of system performance. This study uses GPS data collected from users of various devices at a certain frequency, such as every few seconds. The more often the device transmits its coordinates, the more accurate the GPS track built based on this data will be. All data is anonymized and does not contain any information about a specific user or car. Tracks come not only from private drivers, but also from partner companies (organizations with a large fleet of cars driving around the city). GPS tracks are tied to a road graph, after which the speed of all tracks passing along the same edges is averaged. Such data have a number of features that make it difficult to obtain a high-quality forecast. However, the devices determine the coordinates with an error, so the traces are very noisy. This makes it difficult not only to calculate the average speed, but also to link the track to the schedule, especially when the roads are located close to each other. And the situation on the roads itself is not always stable. For example, highway lanes travel at different speeds, so tracks on the same stretch of road may simultaneously display significantly different speeds. GPS tracks coming from users of smart devices and partner companies need to be associated with a road graph in order to determine as accurately as possible on which streets cars drive or people walk. The speed of all paths passing along the same edges is then averaged. GPS receivers make errors in determining coordinates, which makes it difficult to build a track. The beetle can move an object several meters in any direction, such as onto the sidewalk or the roof of a nearby building. Coordinates received from user devices are transmitted to an information system that accurately displays buildings, parks and streets with road markings and other infrastructure. When processing a stream of raw data, the Viterbi algorithm is used to make the most probable guess about the sequence of states of a hidden Markov model based on a sequence of observations. The Viterbi path represents the most probable sequence of hidden states. The algorithm makes several assumptions:

• observed and hidden events must be a sequence, most often ordered by time;
• the two sequences must be aligned, and each observed event must correspond to exactly one hidden event;
• the computation of the most probable hidden sequence up to time $t$ must depend only on the observed event at time $t$ and the most probable sequence up to time $t-1$.

Using the Viterbi algorithm, users can determine how the object most likely moved: let there be a hidden Markov model with state space $S=\{s_1,s_2,\dots ,s_K\}$, where $K$ is the number of possible different states of the network. In this case, the states that the network accepts are invisible to observation. Let us denote by $x_t$ the state of the network at time $t$. At the network output at moment $t$, the observable value $y_t\in O=\left\{o_1,o_2,\dots ,o_N\right\}$ appears, where $N$ is the number of possible different observable values at the output. Let ${\pi }_i$ be the initial probability of the network being in state $i$ and $a_{i,j}$ be the probability of the network transitioning from state $i$ to state $j$. Let the sequence $y_1,\dots ,y_T$ be observed at the network output. Then, the most probable sequence of network states $x_1,\dots ,x_T$ for the observed sequence can be determined using the following recurrence relations (1):

$$\begin{gathered} V_{1,k}=P\left(y_1|k\right)·{\pi }_k \\ {V}_{t,k}=ma{x}_{x\in S}\left(P\right(y_t\left|k\right)·a_{x,k}·V_{t-1,x}), \end{gathered}$$

(1)

where $V_{t,k}$ is the probability of the most probable sequence of states corresponding to the first $t$ observed values, ending in state $k$. The Viterbi path can be found using pointers that remember which state x appeared in the second equation. Let $Ptr(k,t)$ be a function that returns the value $x$ used to calculate $V_{t,k}$ if $t>1$ or $k$ if $t=1$, then (2):

$$\begin{gathered} x_T=argma{x}_{x\in S}\left(V_{T,x}\right) \\ {x}_{t-1}=Ptr\left(x_t,t\right) \end{gathered}$$

(2)

Extrapolating the Viterbi algorithm to the task of processing GPS tracks, we can come to the following: the received coordinates from the devices must be superimposed on the road graph; for example, a car could not enter the oncoming lane, and a turn must be carried out according to road markings. Taking into account information about the speed of its passage, a single route of movement is constructed—the path, shown in Figure 4.

Each route has a control time during which one can drive on a free road without breaking the rules. Each intersection is a vertex of the graph, and sections of the road between two intersections are edges. The latter have the attributes of length and speed. The length is known in advance and the speed is calculated in real time. After estimating the load on all tracks on the graph, the model calculates how much the real time differs from the reference time. Network congestion is presented on a scale of 0 to 10, where 0 equals free movement and 10 equals traffic stops. The scale is configured differently for each city and has minimum and maximum coefficients, selected empirically, that take into account seasonality and climatic parameters. With this estimate, the decision maker will be able to understand how long it will take to complete the route. For example, if the average traffic score is 7, then the trip will take approximately twice as long as with free traffic. The input parameters of the model are as follows:

• estimated average speed on the road—average speed along the edge; source—OSM data, km/h;
• actual average speed on the road—median speed on a section at a certain point in time; source—data processing model, km/h.

Knowing the calculated and actual speeds, one can calculate the difference between them as a percentage of the planned one (3):

$$∆=(spee{d}_e-spee{d}_a)/\left(spee{d}_e\right)\times 100\%,$$

(3)

where $spee{d}_e$—estimated average speed, $spee{d}_a$—actual average speed.

When searching for a route in navigation applications, algorithms are used to find the shortest path on a road graph. It should be noted that the scientific and practical significance of universal routing algorithms for network models is steadily increasing. This is due to the fact that the formation of ship routes converging at nodes and operational spaces, on approaches to ports, during pilotage and during high and low tide conditions requires the adoption of special measures to ensure the safety and trouble-free operation of ships. Moreover, a systematic solution to these problems will require the development of optimization algorithms based on network models and the creation of a tool base for the robotic unmanned control of ships. Digitalization and development of systems for the intelligent automation of technological processes at transport facilities using neural networks, genetic algorithms and also fuzzy logic systems have found application in models and decision-making algorithms for controlling moving objects in multidimensional operational spaces. Numerous algorithms have been created for modeling and routing on flow network graphs, which actually form tools for finding the shortest paths. Fundamental algorithms include Dijkstra’s algorithms, linear and integer programming and Bellman–Ford and Floyd-Warshall algorithms. The algorithms used to solve optimization problems are based on the general principle of dynamic programming, called the optimality principle by R. Bellman. According to this principle, the solution to the main optimization problem can be the result of optimal solutions to its subproblems, obtained recursively at each computational step [42]. The most popular algorithms for finding the shortest path in a graph are as follows:

• Dijkstra’s algorithm;
• Bellman–Ford algorithm;
• A* algorithm;
• Floyd–Warshall algorithm;
• Lee’s algorithm (wave algorithm).

Let us make a short review of all algorithms.

(1) Dijkstra’s Algorithm:

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Dijkstra’s algorithm is a popular algorithm for finding the shortest path in a graph from a single source vertex to all other vertices.

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It starts by initializing the distance of the source vertex as zero and all other vertices as infinity.

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Then, it iteratively selects the vertex with the minimum distance and updates the distances to its adjacent vertices.

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This process continues until all vertices have been visited or the destination vertex has been reached.

(2) Bellman–Ford Algorithm:

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The Bellman–Ford algorithm is used to find the shortest path between a source vertex and all other vertices in a graph.

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It allows for negative edge weights but detects negative cycles in the graph.

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The algorithm initializes all distances as infinity and the distance of the source vertex as zero.

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Then, it iteratively relaxes the edges by updating the distances until no more updates are possible or a negative cycle is detected.

(3) A* Algorithm:

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The A* algorithm is a popular heuristic algorithm used for finding the shortest path in a graph.

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It uses a combination of Dijkstra’s algorithm and heuristics to efficiently search for the goal node.

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A* estimates the total cost from the start node to the goal node by considering both the cost-so-far and a heuristic function.

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The algorithm uses a priority queue to select the most promising node based on the total cost.

As in-depth explanation of the A* algorithm is as follows:

• Initialize an open set with the start node and a closed set as an empty set.
• Set the value of the initial cost-so-far of the start node as 0.
• While the open set is not empty:

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  Pick the node with the lowest total cost from the open set.

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  If the current node is the goal node, we have found the shortest path.

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  Move the current node from the open set to the closed set.

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  Expand the neighbors of the current node.

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  For each neighbor, calculate the cost-so-far and the estimated total cost.

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  If the neighbor is already in the open set and the new cost-so-far is lower, update its cost.

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  If the neighbor is not in the open set, add it with the new cost.

• If the open set becomes empty and the goal node has not been reached, there is no path from the start to the goal.

Unlike Dijkstra’s algorithm, A* uses a heuristic function to estimate the remaining cost to reach the goal node.

The heuristic function provides an admissible and consistent estimation of the cost.

Admissibility means that the heuristic never overestimates the actual cost, and consistency means that the estimated cost from one node to another is always less than or equal to the actual cost.

(4) Floyd–Warshall Algorithm:

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The Floyd–Warshall algorithm is used to find the shortest paths between all pairs of vertices in a weighted graph.

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It works by iteratively updating the shortest path distances between every pair of vertices using dynamic programming.

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The algorithm maintains a distance matrix that stores the shortest distances between pairs of vertices.

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It iterates through all vertices and considers all other vertices as intermediate vertices to update the shortest path distances.

(5) Lee’s Algorithm:

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Lee’s algorithm is used to find the shortest path in a maze or grid-based graph.

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It performs a breadth-first search from the start node and explores all possible paths until the destination node is reached.

−

The algorithm assigns each cell a unique number representing the number of steps required to reach that cell from the start node.

The most interesting algorithm is A*. Algorithm A* is one of the algorithms for finding the shortest route on a directed graph. Essentially, this is an extension of the famous Dijkstra algorithm. The main difference is that Dijkstra’s algorithm only considers the closest vertices around the start until the wave of detected vertices reaches the end. Dijkstra’s algorithm, being a greedy algorithm, builds the shortest path but does not always do it in the most efficient way. It processes all vertices in the graph, which can lead to unnecessary resource usage, especially in large graphs. The A* algorithm, using a heuristic function, traverses only those vertices that are expected to lead to the goal, which can significantly reduce the number of vertices processed and improve performance, especially in large graphs. Algorithm A* takes into account where the finish is and looks at vertices in the graph predominantly in its direction, which allows it to open, on average, fewer vertices to find the shortest path. When working with A*, the wave of opening peaks goes from beginning to end. There are a few subtle points here. Firstly, in the case of a direct wave (launched from beginning to end), outgoing arcs are sought to open vertices, and in the case of a backward wave (from end to beginning), it is necessary to search for arcs entering vertices to open vertices, which is necessary to preserve the symmetry of the algorithm. Secondly, for a graph, it is advisable to write tests for functions that return functions originating from the vertex and entering the vertex of the arc— this is an important check for correctness. Forward and backward waves can be calculated almost independently until they converge. If users have a processor with two or more cores, they can speed up routing. The proof of correctness must take into account the entire range of possible operation of the algorithm. For example, all the work will be performed using a forward or backward wave, or the waves may converge somewhere in the middle, which is most likely. How a two-way A* will work in two threads depends on the allocated CPU time resources. The important point is that the point at which the forward and backward waves of bilateral A* converge may not be the point of the shortest route. Based on the algorithm, the shortest path from the start to the vanishing point of the waves is calculated, and the path from this point to the finish is also the shortest. However, the shortest path from start to finish can bypass the vanishing point of the waves. To test the efficiency of parallel two-way A*, two smartphones were used: an iPhone X (six-core processor) and a Samsung Galaxy S20 (octa-core processor). Results were obtained for laying out a long route: Rome, Italy to Milan, Italy. Layout time is the average time for laying out a route in milliseconds.

Table 3. Results of two-way A* on different streams.

Method	Samsung Galaxy S20	iPhone X
Double-sided A* in one stream	12,593 ms	20,192 ms
Double-sided A* in two streams	8345 ms	11,459 ms

shows the results of the experiment. Results were obtained directly through time estimation for each method realization.

When implementing the two-way A* algorithm on smartphones like Samsung Galaxy S20 and iPhone X, it is essential to consider the device’s capabilities and optimize the implementation for efficient performance. Here are key features to consider during implementation: memory, computational resources, etc.

Using multiple threads for routing on the Samsung Galaxy S20 gave greater speedup than on the iPhone X. If you need to increase the number of responses per second (RPS), and the fastest response time can be sacrificed, then it makes sense to use two-way A* in one thread so as not to waste resources on synchronization when the algorithm is running. If it is important to speed up routing as much as possible, then it makes sense to parallelize two-way A* into two threads. Tests show that, with a reasonable approach to synchronization, this allows you to gain tens of percents by loading two cores during route calculation.

The problem of assessing and forecasting weather-on-route conditions is addressed using the values of temperature, barometric pressure, relative humidity, wind speed, gusts, cloud cover and visibility measured by a stationary weather station over the previous hour to predict the presence or absence of precipitation for short-term forecasting. The artificial neural network model is based on one of the neural network architectures called multilayer perceptron, with a three-layer structure (one input layer, one hidden layer and one output layer) with activation function $ReLU\left(x\right)=max(0,x)$. on the hidden layer and $Softmax$ on the output layer. The $ReLU$ function returns $0$ if it takes a negative argument, but if it takes a positive argument, the function returns the number itself. $Softmax$ is typically applied to the output layer of multiple classification problems to represent the probability distribution of $K$ categories. The $Softmax$ function takes as its input a vector $z$ of $K$ real numbers and normalizes it into a probability distribution consisting of $K$ probabilities proportional to the exponents of the input numbers. After applying $Softmax$, each component will be in the interval $(0,1)$ and the sum of the components will be equal to $1$, so they can be interpreted as probabilities. $Softmax$ function (4) is as follows:

$$\sigma =\frac{e^{z_i}}{\sum _{j=1}^K{e}^{z_j}}$$

(4)

where $i=1,\dots ,K$ and $z=(z_1,\dots ,z_K)\in R^K$. The network was trained for a fixed number of $Epochs=100$, which shows how many times the model is exposed to training. In the neural network, the objective function is optimized with the selection of weighting coefficients using the adaptive inertia method Adaptive Moment Estimation (Adam)—a variant of stochastic gradient descent. The loss function is used to calculate the error between the predicted and actual values. The loss function is Categorical Cross-entropy (CCE) because it is used as a loss function for a multi-class classification model (5):

$$CCE=-\sum _{i=1}^n\left(d_i·\log \left(y_i\right)\right),$$

(5)

where $d_i$ is the true value and $y_i$ is the probability distribution of model predictions. The accuracy of the neural network model is calculated using the Formula (6):

$$A=1-\frac{Q_N}{Q_O},$$

(6)

where $Q_N$ is the number of incorrectly classified samples and $Q_O$ is the total number of samples. During the training process, two quantities are displayed: the loss on the training data and the loss on the testing data. Figure 5 shows a satisfactory network setup.

To select the optimal neural network architecture, a computational experiment was carried out with the choice of the forecast step size $h$ for 3, 6, 12 and 24 h ahead and the test sample sizes of 10, 15, 20 and 30%. A comparative analysis of the accuracy of the model at different time steps showed that, as the step increases, the accuracy of the model decreases. But it is worth noting that the difference in accuracy when forecasting after 3 versus 6 h is very small. Therefore, due to the greater significance of the above model, forecasting is carried out up to 6 h. Based on the results of numerous runs of training, a multilayer perceptron and testing of the resulting mathematical model, the accuracy of the model is set at 0.86. This means that 86% of forecasts for the presence or absence of precipitation are predicted correctly, but the error is 14%. The test sample contains more than 10,000 observations within a period of 1 h. Of these, 82% are without precipitation and 18% are with precipitation. The number of forecast observations without precipitation with a probability of more than 0.5 is 98%, and the number of forecast observations with precipitation with a probability of more than 0.5 is 35%.

The mathematical models described in the current section will be used as elements of the ranking system described below.

4. Multicriteria Analysis of Decision Support System Parameters

Decision support systems are also important [47]. Based on the predicted indicators, in our case it is possible to predict the optimal redirection of flows, safe or short paths, etc. In this article, authors use a hybrid MCDM system for data analysis. First, weights are assigned to the selection criteria using the AHP method. TOPSIS is used to determine route priority based on normalized weighted criteria. The full list of criteria was obtained through a literature review and questionnaire completion by urban transport planners. A total of 20 experts provided their opinions, which improved the quality of the study. Ordered weights and indices of discrepancy between these weights, taking into account the subjectivity of judgments, were assessed in a survey of a certain format. One of the critical issues in using multicriteria methods is the need to assign weights to the criteria. There are a wide variety of approximation methods used to accomplish this assignment, the most famous of which is AHP. Another requirement of MCDA methods is that of assigning values to criteria, which is subjective in most cases. Information that has been obtained through technical devices is often considered more reliable than information that has been estimated, interpolated or simply interpreted. For this reason, the method of obtaining information related to the criteria plays an important role. In this article, an MCDM approach was used to rank roads and determine their optimality coefficients. The optimality coefficient of a road graph means the relative percentage of proximity of this road (alternative) to the ideal situation—proximity to the ideal criteria chosen by the decision maker. At the first stage, the most important criteria are identified, which are classified into two groups of quantitative and qualitative types. In the next step, data corresponding to each of these criteria is collected from the relevant organizations. After collecting data to determine the weight and importance of each criterion, the criteria are divided into main groups, most of which have sub-criteria. The weight of each criterion is then determined using the GAHP method. After determining the weights of the criteria and their corresponding values for each road, the next step is to rank the roads and determine the optimality coefficients using the TOPSIS method. Quantitative criteria for choosing the optimal path are as follows:

• Distance—affects the travel time and the amount of fuel consumed;
• Travel time—traveling route without taking into account traffic congestion;
• Congestion—density of traffic jams;
• Travel costs—availability of toll roads along the route;
• Safety—dangerous sections of the route affect the optimality of the trip.

Qualitative criteria are criteria that relate to features or objects of the road. The criteria are considered qualitative because their conditions are determined subjectively based on expert opinions. The higher the criterion value, the higher the level of road safety and the lower the likelihood of accidents on the road. Among them, the road type criterion is divided into: highway, main road, secondary road and country road. The criterion for road geometric performance refers to factors such as curve radius calculation, roadway width, road shoulder and excavation slope. Qualitative criteria for choosing the optimal route include the following:

• Type of road;
• Geometric characteristics;
• Quality of road surface;
• Weather;
• Availability of up-to-date data on the route.

A hierarchical diagram of the criteria influencing the choice of route is presented in Figure 6.

The data required to rank and determine the rationality of alternatives (roads) according to quantitative and qualitative criteria were obtained using OSINT and refer to a 24-month period (1 January 2021–31 December 2022). However, for more accurate traffic analysis and assessment, it is recommended to accumulate and consider a longer period. This goal will be achieved through the creation and development of comprehensive databases that store all the necessary statistics on all criteria, without exception, for the qualitative and quantitative characteristics of road traffic events. Data for ranking when choosing the optimal route according to quantitative and qualitative criteria during the study period were for roads in Moscow, Russia. The opinions of traffic experts from relevant organizations were used to assign values to the quality criteria. For this purpose, a subjective interval scale with nine answer options was used. Since all quality criteria are positive, higher values are indicative of a better condition of the criteria. Regarding the road type criterion, it should be noted that each road belongs to one type of this category and has its own meaning. For example, if the roads are highways, they are assigned a value of 9 even if they are not the same in terms of quality and comfort, and the quality of the road is examined according to the criterion of geometric characteristics. The relative weights of the criteria are calculated by normalizing each pairwise comparison combination matrix and taking the arithmetic mean of each row. The weights of the main criteria and sub-criteria, as well as compatibility coefficients, were obtained as a result of the pairwise comparison of combination matrices. Since the coefficient of agreement of all pairwise comparison matrices are less than 0.1, the pairwise comparisons performed are valid and the weights resulting from these comparisons have the necessary validity. The congestion and tariff criteria were found to be the most important and least important criteria compared to others.

Table 4. Main criteria weights.

Priority	Criteria	Weight
1	Road Traffic	0.197
2	Travel Time	0.151
3	Distance	0.122
4	Safety	0.110
5	Weather	0.103
6	Geometric Characteristics	0.101
7	Availability up-to-date information on route	0.077
8	Road Surface Quality	0.063
9	Road Type	0.055
10	Travel Costs	0.021
Weight Sum		1
CR		0.075

presents estimated weights for different criteria.

All further calculations were presented as part of an experiment to find a rational route from point A to point B on a weekday in Moscow, Russia. The shortest route search module contains the calculation of several optimal routes in terms of distance and travel time. When searching for routes, a two-way A* algorithm is used, the input of which is the coordinates of the starting and ending points and road graph data from OSM, and the output of which is alternatives (routes), taking into account a coefficient that differs for urban and intercity trips. The coefficients were obtained empirically, and for city trips they are no more than 80% of the length of the shortest path, and for intercity trips they are no more than 40% of the length of the shortest path. The odds can be changed at any time by the decision maker. The result is shown in Figure 7.

Other criteria were calculated using mathematical models of route congestion, weather definitions described earlier and open sources of data such as Traffic accident database, OpenStreetMap, etc. After calculating, the results are processed using a paired comparison model taking into account priorities received from the decision maker. The result consists of routes calculated in order of least to greatest priority. The final decision on the chosen route remains with the user.

By determining the weight of the criteria, users can rank the roads. For this purpose, multicriteria decision analysis methods, such as TOPSIS, are used. The TOPSIS technique is based on the idea that the selected alternative should be the shortest distance from the positive ideal solution and the greatest distance from the negative ideal solution, which can be expressed in the following steps:

• Calculation of the normalized decision matrix $\left(N\right)$, where the normalized value $N_{ij}$ (7):

  $$N_{ij}=\frac{X_{ij}}{\sqrt{\sum _{j=1}^n{X}_{ij}^2}},i=1,\dots ,m;j=1,\dots ,n$$

  (7)
• Calculation of the weighted normalized decision matrix $\left(V\right)$, where the weighted normalized value $V_{ij}$ (8):

  $$V_{ij}=W_i{N}_{ij},i=1,\dots ,m;j=1,\dots ,n$$

  (8)

  where $W_i$ is the weight of the $i$-th criterion and $\sum _{i=1}^m{W}_i=1$.
• Definition of a positive ideal and negative ideal solution. The positive ideal solution $\left(A^{+}\right)$ for positive criteria includes the highest values of the criterion, and for negative criteria includes the lowest values of the criterion. In addition, the ideal negative solution $\left(A^{-}\right)$ for the positive and negative criteria includes the smallest and largest values of the criterion, respectively (9):

  $$\begin{gathered} A^{+}=\left\{V_1^{+},\dots ,V_n^{+}\right\},\mathrm{where}{V}_j^{+}=\left\{\begin{array}{c}{Max}_i{V}_{ij},ifj\in K\\ {Min}_i{V}_{ij},ifj\in K^{\prime }\end{array}\right. \\ {A}^{-}=\left\{V_1^{-},\dots ,V_n^{-}\right\},\mathrm{where}{V}_j^{-}=\left\{\begin{array}{c}{Min}_i{V}_{ij},ifj\in K\\ {Max}_i{V}_{ij},ifj\in K^{\prime }\end{array}\right. \end{gathered}$$

  (9)

  where $i\in \{1,2,\dots ,m\}$, $j\in \{1,2,\dots ,n\}$ and $K$ correspond to positive criteria, $K^{\prime }$ correspond to negative criteria.
• Calculation of separation measures using $n$-dimensional Euclidean distance. Separating each alternative from a positive ideal and a negative ideal solution (10):

  $$\begin{gathered} d_j^{+}={\left\{\sum _{i=1}^m{\left(V_{ij}-V_i^{+}\right)}^2\right\}}^{\frac{1}{2}},j=1,...,n \\ {d}_j^{-}={\left\{\sum _{i=1}^m{\left(V_{ij}-V_i^{-}\right)}^2\right\}}^{\frac{1}{2}},j=1,...,n \end{gathered}$$

  (10)
• Calculation of relative proximity to the ideal solution. The relative proximity of alternative $A_j$ with respect to $A^{+}$ is defined as (11):

  $$R_j=\frac{d_j^{-}}{\left(d_j^{+}+d_j^{-}\right)},j=1,\dots ,n$$

  (11)

  where $d_j^{+}\ge 0$ and $d_j^{-}\ge 0$, as well as $R_j\in [0,1]$.
• The ranking of the order of alternatives is based on the values of $R_j$. Every alternative with larger $R_j$ is better than other alternatives.

The assessment and ranking of roads were carried out on the basis of open data obtained from the results of the models described above and from open data sources such as OSM. The relative cost of each road was first determined and normalized based on the sub-criteria, then by combining the normalized values of the sub-criteria, the normalized values of their respective main criteria were obtained and used for ranking and scoring besides the other main criteria. To combine the normalized values of the sub-criteria, the relative weight of each was multiplied by the importance value and then summed together. The results are shown in

Table 5. Normalized matrix of main criteria.

Criteria Type		Negative	Negative	Negative	Positive	Positive	Positive	Positive	Positive	Positive	Negative
	Criteria	Road Traffic	Travel Time	Distance	Safety	Weather	Geometric Characteristics	Availability Up-to-Date Information	Road Surface Quality	Road Type	Travel Costs
Route
1		0.252	0.270	0.118	0.139	0.319	0.305	0.334	0.270	0.367	0.309
2		0.109	0.169	0.263	0.184	0.144	0.092	0.110	0.384	0.326	0.184
3		0.268	0.288	0.202	0.393	0.342	0.152	0.186	0.173	0.169	0.384
4		0.200	0.179	0.207	0.266	0.316	0.283	0.252	0.309	0.242	0.282

.

After obtaining the normalized decision matrix, the weighted normalized matrix was obtained by multiplying this matrix by the diametric matrix of criterion weights, and on its basis the ideal positive $A^{+}$ and negative $A^{-}$ solutions were determined.

The results show in

Table 6. Positive and negative ideal solutions.

Positive Ideal Solution (A+)	Negative Ideal Solution (A−)
0.045	0.051
0.086	0.033
0.009	0.063
0.041	0.084
0.071	0.144
0.022	0.092
0.029	0.11
0.065	0.072
0.197	0.026
0.028	0.053

.

At the next stage, the distances of each alternative (route) to the positive $d_j^{+}$ and negative $d_j^{-}$ ideal solutions were obtained, on the basis of which relative proximity was achieved in relation to the ideal solution $R_j$. When planning a route from point A to point B, it is possible to plot five optimal routes taking into account the basic criteria of distance and travel time. It is important to note that possible routes are composite and consist of several elements, so when calculating the rationality of a route, it is necessary to calculate each element of the composite route, sum it up and get the final result.

Table 7. Optimality coefficients of alternatives.

Alternative	${\mathit{d}}_{\mathit{j}}^{+}$	${\mathit{d}}_{\mathit{j}}^{-}$	${\mathit{R}}_{\mathit{j}}$	Route Optimality Coefficient	Optimality Share Percentage
1	0.037	0.160	0.813	81.3	28.67%
2	0.103	0.084	0.451	45.1	15.90%
3	0.060	0.171	0.740	74.0	26.09%
4	0.032	0.158	0.832	83.2	29.34%

presents optimal coefficients.

Based on the calculation results, the most optimal route is Alternative 4, shown in Figure 8. Obviously, with improved conditions, the use of the opinions of more experts and the availability of more historical data, it is possible to make a more accurate and complete assessment of the optimality of each route. By increasing the duration of the study period and updating the criteria values, a better analysis can be performed.

5. Computer Vision Traffic Monitoring

Congestion estimation based on computer vision [48] is an important task. Operational information about the number of cars at different points and prediction of their movement vectors will allow the timely modification of decisions issued by the support system based on multicriteria analysis and mathematical modeling. And in [49], computer vision is even used to detect various incidents on the road. It can also be an important factor for re-evaluating, for example, route safety criteria.

So, in traffic control, an important factor is the possibility of real-time traffic estimation, providing the necessary adjustments to the graph structures due to the current traffic congestion. Such tasks were successfully solved in [44] using convolutional neural networks. However, it was necessary to improve the accuracy of vehicle recognition, especially for small-sized objects. Therefore, a transform-based architecture was chosen [50,51]. Thus, in this section, the optimization algorithms of the transformer model are investigated and the comparison is made with the latest version of the YOLOv8 single-pass detector [52].

It should be noted that the literature review revealed several similar solutions. For example, in [53], the authors detected vehicles using an attention mechanism model. Despite the fact that the authors claimed that the algorithm was a performance algorithm, the model used is YOLOv5, which is slower than newer versions. The work of [54] confirms the good quality of the YOLOv8 algorithm, but it did not optimize the system for higher performance computations. And in [55], the architecture was modified based on U-Net like networks to efficiently process satellite images for which dimensionality reduction is needed.

A set of images with small-sized cars was used, namely 2100 images and more than 15,000 vehicles were prepared for comparison. The partitioning was performed in the Roboflow system. Model training was performed on an NVIDIA RTX 2080 Super GPU graphics card. Both algorithms used data augmentation techniques presented in [56,57]. It should be noted that augmentations were used exactly at the level of feeding images to the neural network input, which allowed a significant increase in the variability of the training sample, in contrast to the method for generating augmented images in advance, as proposed by Roboflow.

Advanced hyperparameter settings such as the ADAM optimization algorithm and a learning rate with scheduler from 0.001 to 0.0001 were used in training. Also, the sample was divided into training, validation and test samples in proportions of 70-20-10%. In addition, the preprocessing steps built into Roboflow were used, namely resizing to a fixed 640 by 640, and normalization of the luminance data. Selection of the number of images in the batches was performed using grid search. The optimal value was eight images.

In addition, ADAGRAD, RMSProp and SGD optimization algorithms were also tested, but their results were inferior to those of ADAM. Therefore, the subsequent models were built using ADAM as the optimization algorithm. The use of the ADAM optimization algorithm has shown better performance in terms of convergence and achieving higher accuracy during training. This emphasizes the importance of choosing the appropriate optimization algorithm for the specific task and dataset. Overall, a thorough exploration of hyperparameters and optimization methods, combined with proper preprocessing and data splitting, has contributed to the building of high-performing models based on the ADAM optimization algorithm for the specific task at hand.

The target metric for comparing the models in terms of quality was the mAP Average Precision metric, while the frame per second (FPS) metric, representing the actual average number of frames processed per second, was chosen to evaluate speed.

The efficiency metric mAP (mean Average Precision) is a metric used to evaluate the performance of object detection and recognition algorithms. It measures how well the algorithm can accurately locate and classify objects within an image or a set of images.

In simple terms, mAP tells us how well an object detection algorithm can find and classify objects in images, considering both the accuracy of the predictions and the ability to detect different types of objects. A higher mAP indicates better performance, meaning that the algorithm is able to locate and classify objects more accurately and consistently across different categories.

Table 8. Baseline models for small vehicle detection.

Model	Efficiency, mAP	Performance, FPS
YOLOv8x	0.901	11.62
DETR	0.934	7.23

presents the results from comparing the YOLO and DETR models on the test sample.

The analysis of the results presented in Table 5 shows that the transformer neural network for computer vision has a gain of about 3% in accuracy but is inferior by about 1.5 times in speed.

Figure 9 shows the loss functions on the left and the mAP metric during training on the right.

It can be seen that the curves are going on the plate. So, the DETR model is a little better than the YOLO model.

Next, optical flow technology [58] was used. Its application provided acceleration of the DETR model up to 12.92 FPS without quality loss. Figure 10 shows an example of image processing from a multi-camera stream. So, we called the new optimized version of neural network detectors a productive model, as it provides better performance.

It can be seen from Figure 10 that successful image matching and effective detection of all objects of interest is accomplished.

Figure 11 shows a block diagram of the processing. The main contribution of this paper is that, based on the optical flow, the future position of the object in the frame is predicted, due to which, the desired region of interest is used in detection. At some periodicity (T = 3), the detector is completely restarted.

So, the novelty of our contribution is in the joint use of a neural network detector with optical flow velocity orientation and a speed predictor. The main advantage of such a solution is an increase in processing speed, but the main disadvantage is the accumulated error in the optical flow algorithm. To solve this, once every three frames, the detector performs a detection from zero, which also resets the accumulated error to zero.

Thus, a high accuracy and high-performance computer vision system for congestion estimation and traffic prediction is proposed.

Finally, a predictive time-series model was built based on the dynamic data to determine the number of vehicles on the road section. It is proposed to use CNN+LSTM architecture, and the comparison is made with RNN and GRU networks, as well as with ARIMA and AR statistical models.

In the context of traffic monitoring and management, recurrent algorithms play a crucial role in processing sequences of data, leading to enhanced accuracy in traffic estimates. Recurrent algorithms are specifically designed to handle sequential or time-series data, making them suitable for analyzing traffic patterns over time.

One of the key advantages of recurrent algorithms is their ability to capture temporal dependencies and contextual information. Traditional algorithms, such as linear regression or decision trees, are not well-suited to handling the inherent time-varying nature of traffic data. Recurrent algorithms, on the other hand, are capable of retaining the memory of past information while processing current inputs, allowing them to capture long-term dependencies and patterns in the data.

For comparison, we used the mean square error (MSE) and mean absolute percentage error (MAPE). The data containing one week of record processing were used.

Table 9. Count of vehicles prediction error.

Model	MSE	MAPE
AR	36.82	26.88
ARIMA	29.12	24.15
RNN	18.76	13.64
GRU	19.22	14.10
Ours	12.65	9.82

provides the prediction results. So, here we used neural network detectors for preprocessing for counting the number of vehicles in different time moments. After that, we worked with a simple time series.

The analysis is Table 9 shows that the proposed model performs better in a traffic prediction task. This is due to the additional consideration of short-term and long-term dependencies.

It should be noted that for all machine learning experiments, the comparison of the estimators’ results with the predictions was performed using the classical method using human-labeled ground truth test data.

6. Conclusions

This article provides an overview of modern mathematical methods for analyzing traffic flows. The algorithm used in building and displaying traffic is described in detail. The results of the work were tested on real anonymous data from user devices. A multicriteria model for finding a rational route has been developed. The coefficients of the multicriteria model were calculated using the OSINT technique. In addition, a rapid vehicle detection algorithm has been developed to monitor and verify the quality of traffic flow. In the future, the analysis data generated by this system can be integrated into traffic route design models taking into account traffic situations.

The quality metric should reflect how much the new method helps users. Based on the results of closed testing, a survey was conducted to determine how well the proposed route satisfies each user’s specified conditions. Thirty users took part in the survey, and their answers were distributed as follows:

• Completely satisfied—10
• Satisfied—11
• Does not satisfy—6
• Completely unsatisfied—3

The system showed acceptable results for the further improvement of algorithms and data processing.

As for computer vision systems and traffic prediction models, the research showed that it is possible to effectively process video data for counting vehicles in real-time and for predicting the number of vehicles in the future. Jointly with computer vision systems, such intelligent models can be used for dynamic route optimization.

The limitations of this algorithm can be technical problems with sensors and cameras, which can be solved by duplication, which will increase the cost. In future research, it is planned to approach the problem of traffic prediction in more detail, since the experiments with recurrent networks in this paper showed a good basic result, but one which can be improved.

Possible contributions in the future are also as follows. The integration of advanced computer vision techniques into the multicriteria assessment method for network structure congestion based on traffic data brings profound enhancements to the evaluation process. By leveraging object detection, crowd density estimation, traffic flow analysis and anomaly detection, the research offers a comprehensive understanding of congestion dynamics within the network structure. This provides stakeholders with valuable insights for making informed decisions and implementing effective congestion management strategies. The results of this research contribute to the advancement of both computer vision and transportation engineering fields, propelling the development of intelligent and efficient transportation systems.